Annual Equivalent Cash Flows

I was working on my finance homework and came across this problem that I could not figure out. if anyone knows how to do this please let me know..A firm needs to decide which machine to purchase. Factory 1 costs $3,000, has a useful life of three years, and has annual maintenance costs of $800. Factory 2 costs $12,000 and has a useful life of four years, and has annual maintenance costs of $1,000. Both are worthless after their useful lives of operation. They are expected to generate the same annual cash flows. The discount rate is 10% for both factories. Based on annual equivalent cash flows, which factory should the company select?

Yep, this one's certainly one of the more obscure corp fin topics. The basic idea is that you can't make a choice using a straight-up NPV comparison, as the assets have unequal lives. So the approach is to convert the numbers into an apples/apples comparison using a (rather rickety) assumption that whichever model you go with initially, you'd stick with it long run by continuing to replace it each time with replicas of itself.

Here's the approach: Model 1 requires a 3-year cash outflow pattern of 3,000; 800; 800; and 800 over its three-year life (I'm assuming the 3K is paid immediately, and the annual maintenance costs are paid at the end of each year). The main idea is that you convert this into an "annual equivalent" cash flow set, in the form of a 3-year annuity.

At 10%, the asset's cash outflows as described have a PV of 4,989. However, a 3-year ordinary annuity of 2,006 per year, 3 years, also has a PV of 4,989. Thus, buying Model 1 now, and replacing it with a new Model 1 every three years, is economically equivalent to paying out a perpetuity of 2,006 at the end of each year from now on.

The 2,006 is Model 1's "annual equivalent cash flow". Now do the same for Model 2, but remember you need to come up with a four-year annuity that has the same PV as the asset itself, since this model would be replaced every four years. Finally, you go with the one that has the lower perpetuity equivalent.

I have to say, though, that in this case, if your book / instructor wasn't telling you to, you wouldn't need to go through this exercise. Model 1's 3K every 3 years is a lot cheaper than 12K every 4 years, and its annual maint costs are lower, to boot. If they do indeed generate the same cash inflows, it's a no-brainer.

(and FWIW, I've never been too wild about the assumption that underpins this method) Cheers, all...

I have to say, though, that in this case, if your book / instructor wasn't telling you to, you wouldn't need to go through this exercise. Model 1's 3K every 3 years is a lot cheaper than 12K every 4 years, and its annual maint costs are lower, to boot. If they do indeed generate the same cash inflows, it's a no-brainer.